As design icons go, it’s hard to beat a Sputnik: for its combination of sphere and line, its double otherworldliness (to westerners), being both cosmic and Soviet, its mixed message of global unity and intercontinental, potentially nuclear, rivalry. It caught a moment, summarised the space age and encouraged a style of spindliness and volume that shaped motels and table lamps.

Like the best icons, it wasn’t designed for effect but for the fearsome expediencies of the race between the USSR and the US to put the first manmade object into orbit, and for the changed physical rules of the previously unknown world outside the Earth’s atmosphere. At the same time it was part of a movement already decades old at the time that the first Sputnik was launched in 1957, which combined fantasy and image with function and science. In the Science Museum’s Cosmonauts exhibition, which covers Russia’s fascination with space from late tsarist times to the present, it is possible to see the country’s extraterrestrial exploration as a vast art project.

Exhibits include early abstract paintings representing new cosmic orders and Konstantin Yuon’s New Planet, a 1921 painting intended for the drop curtain of the Bolshoi theatre, where the October revolution is shown as a previously unknown world, floating amid strange spheres and bursts of radiant light. There is also the 1928 Labour Commune by an architecture student called Georgy Krutikov, a series of circular space stations that formed part of his atomic-powered Flying City project. Space was seen as a place where the revolutionary project of a perfect society could reach fulfilment.

A key figure is Konstantin Tsiolkovsky, a bearded provincial schoolteacher, almost completely deaf, who wielded a vast ear trumpet of his own design and manufacture and who from the late 1870s until his death in 1935 studied questions of weightlessness and space travel, and tried to communicate them through science fiction. The exhibition shows his notes and sketches for a 1930s film called Cosmic Voyage, on which he was a consultant. Their style has a schoolboy naivety, but they show insights into the ways and means of reaching space, including escape velocity, fuel, and the use of multi-stage rockets remarkably close to those that would eventually be employed.

Cosmic Voyage, 1935

Tsiolkovsky and others followed the frankly bonkers philosophy of Cosmism, which proposed immortality and resurrection through science. Yet his ideas of space travel became reality through the efforts of another hero of the exhibition, Sergei Korolev, who despite spending some years in the Gulag drove forward the Soviet programme that put the first object, animal, man and woman into space. At which point exploration became a tool of geopolitical propaganda, giving rise to monuments and celebratory posters.

The most remarkable pieces in this gripping show are the technical equipment, the capsules, landers and probes, not to mention the pressurised trousers, space suits and control tables, that have the profound strangeness that comes from addressing huge and unusual technical challenges. Some of them baked by the heat of re-entry, they can look strikingly primitive as well as technologically wondrous. Their weirdness is beyond anything that the wildest dreamers could have conceived. At the same time the art and design that inspired the space programme and was inspired by it, from 1920s abstraction to Sputnik-shaped souvenir samovars, shows how science is anything but pure, but is embedded in the societies from which it comes.

One thing beyond the Cosmists’ fertile imagination would have been that, in the 21st century, considerable care would have been taken to restore a wrecked relic of the 19th to regular use, while retaining its character of wrecked-ness. This is what has happened to Wilton’s Music Hall in east London, a rare survivor that flourished in its original use for little more than 20 years, was taken over by Methodists and, since it was listed in 1971, has undertaken a slow journey to a renovation. Performances started there again in the 1990s, but it is only under the leadership of its current director, Frances Mayhew, that it has reached a position where it can deliver a regular programme into a relatively secure future. In 2004 Mayhew rescued it from possible abandonment or, worse, a takeover by Wetherspoons, and has been championing it ever since.

The carefully restored Wilton’s Music Hall, which has retained ‘its character of wrecked-ness’. Photograph: Helene Binet

The complex consists of a row of houses that once contained a pub, which was acquired by the eponymous John Wilton, who than added a purpose-built auditorium at the back where, rather than face the stage in solemn rows, customers could eat and drink and be entertained by performers as much or as little as they desired. The auditorium reopened in 2013; the remaking of the former houses, to include a rambling array of rooms, bars, offices, rehearsal space, and some of open-ended purposes is now complete. It is not precisely going back to those days, but it is reviving a spirit of looseness, in which several ways of using the theatre are possible, and in which the common spaces of the old houses can be inhabited and activated, also in multiple ways. Performance is not confined to a designated zone, but can happen almost anywhere.

The appeal of Wilton’s, ever since Fiona Shaw and Deborah Warner put on TS Eliot’s The Waste Land there in 1997, has been its fragility, the feeling that it might not be there tomorrow, which at the time was almost true. With the latest work, the architect Tim Ronalds has had to spend £4m on ensuring its actual permanence, without losing this vital quality. Crude brickwork, not originally intended to be seen, is therefore left unplastered, as are Victorian lash-ups of reused railway track and cement. There are spaces that seem to have escaped from a Gustave Doré engraving, others that look almost medieval. The auditorium is coated in worn layers of paint, mysterious and apparently venerable, but actually the residue of 1990s performance and film shoots such as Richard Attenborough’s Chaplin.

This ruinenlust would have been beyond the comprehension of John Wilton, for whom the object was to create a palatial, chandelier-lit escape from the grotty reality of its surroundings. The current Wilton’s is different, but it still draws on the aspirations of the original, expressed in its slender barley-sugar columns and ornamental bas-reliefs on the bar. Its magic now comes from the dreaming and the crumbling together: a replication of the original decor would only have been kitsch. It is a construction of multiple layers of fiction, accrued over time. Which, really, is a good thing to have in a theatre.